Self insertion only worked in a few cases for me. With legend of zelda, I always saw Link. The silent protagonist thing never changed that. With Persona 3 and 4, strangely, self insertion worked to a small degree. Though it led me to create more of a separate character based off of me. With skyrim, I completely insert my own character personalities and mindsets over me and focus on telling their story. That to me is more fun than raw self insertion. I guess thats RPC insertion? XD
Meanwhile there have been games where I would have preferred to have self insertion. I dunno, I always thought Mass Effect and The Witcher could have benefited from having a full character customization sequence, but that may just be because they give you so many dialogue options, and yet someone else's character is saying them all. But for the majority of visual novels, I'm fine with that, since the decisions come at pivotal points and not every time you wish to respond to someone asking you if you would like to by a sweetroll with five contrasting options. Just give me a fun, proactive protagonist, and let me enjoy his/her point of view. In fact when I think about it, a voiced protagonist in VNs might actually be preferable, though voiced narration would really be kind of tricky to do right without grating on my nerves. It would require some synthesis on both the writer and the voice actor's part. But voiced dialogue is far more preferable to games that have an actual character as their protagonist, and not just some sort of blank faced "Hi, I'm plain as f*ck. Be me and live out your galge desires!"
Well jeez, my opinions are really all over the place there, aren't the?
To sum it up, I guess:
The more options I'm given in dialogue, the more I prefer a degree of self immersion.
The more freedom I get to explore the world, the more I prefer to insert a character of my own device there.
And for heavy story centric games, with choices only popping up at certain points in the story, I prefer a character with a strong personality to, and I'm there to watch their story. And in this case, I'm perfectly happy with voiced dialogue, but a bit apprehensive on voiced narration. Open, but wary.